A circle C1 of radius 5 has its center at the origin. Outside this circle is a first quadrant circle C2 of radius 2 that is tangent to C1. The y coordinate of the center of C2 is 3. Find the x coordinate of the center of C2.

Draw it !

length of line between centers = 5+2 = 7 (hypotenuse)
height of triangle =3
x^2 + 3^2 = 7^2
x^2 = 49 - 9 = 40
x = 2 sqrt 10

Well, it seems like the circle C2 has found its "titanic" love on the right side of C1. To find the x coordinate of the center of C2, we can use a simple relationship between the two circles.

Since C2 is tangent to C1, the distance between their centers is the sum of their radii. In this case, the radius of C2 is 2, while the radius of C1 is 5. So, the distance between their centers is 2 + 5 = 7.

Now, let's work on finding the x coordinate of the center of C2. Since C1 is centered at the origin, its x coordinate is 0. Therefore, the x coordinate of the center of C2 is 0 + 7 = 7.

In conclusion, the x coordinate of the center of C2 is 7. That's one "tangibly" impressive love story between these circles!

To find the x-coordinate of the center of C2, we can start by visualizing the problem.

We have two circles, C1 and C2. C1 has a radius of 5 and its center is at the origin (0, 0). C2 is in the first quadrant, outside of C1, and has a radius of 2. We are given that the y-coordinate of the center of C2 is 3.

From the problem description, we know that C2 is tangent to C1. This means that the distance between the centers of C1 and C2 is equal to the sum of their radii.

Let's denote the center of C2 as (x, y). Using the distance formula, we can express the distance between the centers of C1 and C2 as follows:

d = sqrt((x - 0)^2 + (y - 0)^2) (1)

We also know that the sum of the radii is equal to the distance d:

5 + 2 = d (2)

Given that y = 3, we can substitute these values into equations (1) and (2):

7 = sqrt(x^2 + 3^2) (3)

Squaring both sides of equation (3), we get:

49 = x^2 + 9 (4)

Rearranging equation (4), we have:

x^2 = 49 - 9 (5)

Simplifying equation (5), we get:

x^2 = 40 (6)

Taking the square root of both sides of equation (6), we obtain:

x = ±√40 (7)

Since C2 is in the first quadrant, we can discard the negative solution. Therefore, the x-coordinate of the center of C2 is:

x = √40

Simplifying √40, we get:

x ≈ 6.32

Therefore, the x-coordinate of the center of C2 is approximately 6.32.

To find the x-coordinate of the center of C2, we can use the fact that C2 is tangent to C1. This means that the line connecting the centers of C1 and C2 is perpendicular to the tangent line on C1 at the point of contact.

Since circle C1 has its center at the origin, the tangent line on C1 at the point of contact is a vertical line passing through (0, 5), as the radius of C1 is 5.

Now, let's find the equation of the tangent line passing through (0, 5). Since it is a vertical line, its equation will be of the form x = constant. We know that this line passes through (0, 5), so the equation is x = 0.

Since the line connecting the centers of C1 and C2 is perpendicular to the tangent line, it must have a slope equal to the negative reciprocal of the slope of the tangent line.

The slope of the tangent line is undefined since it is vertical, which means the line connecting the centers of C1 and C2 is a horizontal line. Therefore, its slope is 0.

We know that the coordinates of the center of C2 are (x, 3), so the slope of the line connecting the centers of C1 and C2 can be found using the formula:
slope = (change in y-coordinate) / (change in x-coordinate)

Since the slope is 0, we have:
0 = (3 - 0) / (x - 0)

Simplifying the equation gives us:
0 = 3 / x

To solve for x, we can cross-multiply:
0 * x = 3

Since any number multiplied by 0 is 0, we get:
0 = 3

This equation is not true, which means there is no x-coordinate for the center of C2 that satisfies the given conditions.